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# Armanios-Wells graph

There is a unique distance-regular graph Γ with intersection array
{5,4,1,1;1,1,4,5}. It has 32 vertices and spectrum
5^{1} 2.236^{8} 1^{10} (-2.236)^{8}
(-3)^{5}.
It is known as the Wells graph, or the Armanios-Wells graph.

It is antipodal of diameter 4,
the unique double cover without 4-cycles of the folded 5-cube.

For each vertex, the subgraph on the vertices at distance 2 is the
dodecahedron. The dodecahedron is itself antipodal, of diameter 5,
and antipodal pairs at distance 5 in the dodecahedron are antipodal
at distance 4 in the Wells graph.

## Group

The full group of automorphisms is 2_{-}^{1+4}:Alt(5)
acting distance-transitively with point stabilizer Alt(5).
## Independence number

The independence number is 10, and the only 10-cocliques
are the lifts of the 5-cocliques in the folded 5-cube.

## Chromatic number

The chromatic number is 4.

## Characterization by spectrum

Van Dam and Haemers constructed by switching
a graph cospectral with the Wells graph, and E. Spence showed by computer
search that there are precisely three graphs with the spectrum of
the Wells graph.

## History

In [BCN] this graph was called the Wells graph.
It was constructed by A. L. Wells (1983),
but also, presumably earlier, by C. Armanios (1981, 1985).

## References

BCN, p. 266.
C. Armanios,
*Symmetric graphs and their automorphism groups*,
Ph.D. Thesis, University of Western Australia, 1981.

C. Armanios,
*A new 5-valent distance transitive graph*,
Ars Combin. **19A** (1985) 77-85.

E. R. van Dam & W. H. Haemers,
*Spectral Characterizations of Some Distance-Regular Graphs*,
J. Algebraic Combin. **15** (2002) 189-202.

A. L. Wells,
*Regular generalized switching classes and related topics*,
Ph.D. Thesis, University of Oxford, 1983.